Substructure and Maximal Common Substructure Searching

Abstract

There are three important and closely related methods for the manipulation and searching of chemical structures: structure isomorphism identification, substructure searching (SSS), and maximal common substructure (MCSS) perception [1]. To understand the internal relationships of these three concepts, it is useful to define several terms before further proceeding with the discussion of SSS and MCSS problems. Because the structure of a chemical molecule can be treated as a mathematical graph with atoms corresponding to vertices and bonds to edges, it is convenient to introduce the following definitions [2] according to graph theory. Definition 1: A graph G= (V, E ) is a finite set of vertices (V ) and a finite set of edges (E ). Each edge (vi, vj) consists of an unordered pair of distinct vertices. Two vertices vi and vj of a graph are said to be adjacent if (vi, vj) is an edge of the graph [3]. If vertices and edges of a graph possess colors, we say it is a colored graph. The colors of vertices and edges of a colored graph separately correspond to the properties of atoms and bonds of a chemical structure. Definition 2: Two graphs G1 = (V1, E1) and G2 = (V2, E2) are said to be isomorphic if their vertices can be identified in a one-to-one fashion so that, if v1i and v1j are vertices inG1, and v2i and v2j are the corresponding vertices inG2, then (v1i, v1j) is an edge ofG1 if and only if (v2i, v2j) is an edge ofG2 [3]. For two colored graphs, the vertex pairs ‘‘v1i, v2i’’ and ‘‘v1j, v2j,’’ and edge pairs (v1i, v1j) and (v2i, v2j) must have the same kind of colors, respectively. Definition 3: Given two graphs G1 = (V1, E1) and G2 = (V2, E2), we say G1 is a subgraph of G2 if V1 is a subset of V2 and E1 is a subset of E2 [3].